Design And Modeling Of Waveguide-coupled Single-mode ...users.eecs.northwestern.edu/~sth/HO JLT...

JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 8, AUGUST 1998 1433

Design and Modeling of Waveguide-CoupledSingle-Mode Microring Resonators

M. K. Chin, Member, IEEE,and S. T. Ho,Member, IEEE

Abstract—We discuss a first-order design tool for waveguide-coupled microring resonators based on an approximate solutionof the wave propagation in a microring waveguide with micron-size radius of curvature and a large lateral index contrast. Themodel makes use of the conformal transformation method and alinear approximation of the refractive index profile, and takes intoaccount the effect of waveguide thickness, dispersion, and diffrac-tion. Based on this model, we develop general design rules for themajor physical characteristics of a waveguide-coupled microringresonator, including the resonance wavelength, the free spectralrange, the coupling ratio, the bending radiation loss and thesubstrate leakage loss. In addition, the physical model providesleads to alternative coupling designs. We present two examples,one using a phase-matching parallel waveguide with a smallerwidth than the ring waveguide, and the other using a verticalcoupling structure. Both these designs significantly increase thecoupling length and reduce or eliminate the dependence on anarrow air gap in a waveguide-coupled microring resonator.

Index Terms—Conformal transforms, microresonators, nano-fabrication, nanophotonics.

I. INTRODUCTION

OPTICAL ring waveguide resonators are useful compo-nents for wavelength filtering, multiplexing, switching

and modulation [1], [2]. The main performance characteristicsof these resonators are the free-spectral range (FSR), thefinesse (or -factor), the transmission at resonance, and theextinction ratio. The major physical characteristics underlyingthese performance criteria are the size of the ring, the propaga-tion loss, and the input and output coupling ratios (equivalentto the reflectivities of a Fabry–Perot resonator). There arevarious components of losses, including sidewall scatteringloss, bending radiation loss, and substrate leakage loss. Ringresonators based on (rib) waveguide structures with weaklateral confinement have very low sidewall scattering loss.However, the high radiative bending loss limits its minimumradius to about 2 mm, in which case the FSR will be small(in the order of 100 GHz [3]), and the finesse will also below because the scattering loss is very high for such a largecircumference, although the scattering loss per cm is lower.On the other hand, strongly guiding ring waveguides withvery large lateral refractive index contrast (air-semiconductor-air) can have diameters as small as 1–2m with negligible

Manuscript received January 7, 1998; revised April 27, 1998. This workwas supported by AFOSR/ARPA under Contract F49620-1-0262 and by theNSF under Grant ECS-9218494. This work was also supported in part byUnited States Integrated Optics, Inc.

The authors are with the Department of Electrical and Computer Engineer-ing, McCormick School of Engineering, Northwestern University, Evanston,IL 60208 USA.

Publisher Item Identifier S 0733-8724(98)05661-8.

Fig. 1. A schematic of the waveguide-coupled microcavity resonator, show-ing a microring resonator coupled to straight waveguides.

bending loss. Therefore, compared with the weakly guidingwaveguide, the strongly guiding microcavity can be up to 1000times smaller. Because of the small size, the propagation lossin the cavity is also negligible. Consequently, both the FSRand the finesse are much larger for this resonator. An FSR of6 THz (50 nm) can be achieved, sufficiently large to cover theentire 30-nm erbium amplifier bandwidth.

Fig. 1 shows a schematic of an add–drop filter (or opticalswitch) implemented with a microring resonator [4]. Thetwo tangential straight waveguides (WG1 and WG2) serveas evanescent wave input and output couplers. If the signalentering port A is on-resonance with the ring or disk, then thatsignal couples into the cavity from WG1, couples out from thecavity into WG2, and exits the device at port C. A signal thatis off-resonance with the cavity remains in WG1 and exits atport B. The critical dimension in this optical switch is the 0.1-

m gap separating the microring cavity from the tangentialwaveguides. These gap sizes determine the input and outputcoupling ratios of the resonator, which in turn determine themagnitude of the finesse and the at-resonance transmittance.In the case of a microring coupled to straight waveguides, thegaps are very small due to the strong optical confinement andthe small coupling interaction length. These small gaps aredifficult to fabricate. In particular, it is difficult to ensure thatthe two coupling gaps on both sides of the ring are identical.If the coupling factors are not equal, then the finesse and theON–OFF ratio of the resonator will be reduced. To achievemore dependable performance, it is, therefore, useful to designalternative coupling structures that significantly reduce thedependence on a narrow air gap.

The aim of this paper is to model the wave propagationin a microring waveguide, to develop a first-order designtool that provides guidance for alternative coupling designs,

0733–8724/98$10.00 1998 IEEE

1434 JOURNAL OF LIGHTWAVE TECHNOLOGY, VOL. 16, NO. 8, AUGUST 1998

and to formulate general design rules for the major physicalcharacteristics of a waveguide-coupled microring resonator.While there are available very precise numerical methods,such as the finite difference time domain (FDTD) method [5],that can solve the electromagnetic field distribution in anycomplicated three-dimensional (3-D) photonic structures, theyare time consuming to use and, for most purposes, the accu-racy is not essential for first-run designs. Recently, multiplecoupled-resonator filters have also been proposed and modeledusing the so-called “coupling of modes in time” method, inwhich the ring eigenmodes are never explicitly solved [6]. Bycontrast, we will follow the more conventional approach inwhich the waveguide eigenmodes are first solved, and thenused to calculate the other parameters such as the couplingcoefficient and radiation loss. This will be achieved usinga model that is relatively simple and intuitive, approximatebut reasonably accurate, and analytical wherever possible. Theadvantage of such an approach, as will be shown, is the gain inphysical insight. Some approximations are necessary in orderto simplify computations and the complexity associated withthe 3-D nature of the problem. In particular, the treatment ofthe vertical dimension (thickness) of the device is approximate.Nevertheless, the model attempts to account for some ofthe 3-D physical effects such as waveguide dispersion, edgediffraction, and bending loss.

II. M ETHOD OF CALCULATION

AND PHYSICAL CONSIDERATIONS

The current method is an extension of the conformal trans-formation method used previously for solving the waveguidemodes of a microdisk structure [7]. This method is chosen be-cause of the intuitive physical picture it gives, as will becomeapparent later. We consider a semiconductor waveguide withcore thickness , and core material index , surrounded inthe horizontal direction by air. The cladding material index is

, which for simplicity is assumed to be the same for boththe upper and lower cladding layers. Two possible realizationsfor the vertical layer structure are shown in Fig. 2(a) and(b) where Fig. 2(a) is a weak vertical guiding structure with

. Since photons are strongly confined only in thehorizontal direction, we call this theType I or the photonicwell structure. Fig. 2(b) is a strong vertical guiding structurewhere . (For example, the low-index medium couldbe air, or acrylic/polyimide with .) Since the photonsare strongly confined in both transverse dimensions we callthis theType II or the photonic wirestructure. In theory thelater structure is more interesting as photons are more stronglyconfined, but in practice it is not easy to realize, especially forelectrically contacted devices.

A. Conformal Transformation Method for RingWaveguides with Finite Thickness

In cylindrical coordinates, the ring waveguide mode may begiven by the 3-D scalar wave equation

(1)

(a) (b)

Fig. 2. Two possible realizations of the vertical layer structure: (a) Type Ior weak vertical guiding structure and (b) Type II or strong vertical guidingstructure.d is the waveguide core thickness andt2 is the lower cladding orbuffer thickness.

where, and is the free space wavelength. For each

vector there are two modes corresponding to the two possiblepolarizations. These are the transverse electric (TE) and thetransverse magnetic (TM) modes with the electric field andthe magnetic field, respectively, parallel to thedirection.For simplicity we will consider only the TE mode, since themethod of analysis is similar for the TM mode. For the TEmode, is the component of the magnetic field (i.e., theelectric field has no component).

The conformal transformation to be discussed later appliesonly to the - plane. We, therefore, first reduce the 3-D waveguide problem to two-dimensional (2-D) by makingtwo approximations. In the first approximation, we assumethat the waveguide mode can be written in the form

, where describes the mode in the- plane and describes the resonant field in the

direction. In the second approximation, we use the effectiveindex method (EIM) [8] to determine and its associatedwavevector component

(2)

For verification of this approximation, we compare the effec-tive indexes for both Type I and Type II waveguidescalculated with EIM and with multigrid finite-difference nu-merical methods [9]. For example, the difference inbetween the two methods is shown for TMmode in Fig. 3.Note that the EIM approximation is a good one except whenthe width is smaller than 0.2m for Type I waveguides and0.4 m for Type II waveguides.

Fig. 4(a) shows the approximate for the TE modes as afunction of the normalized core thickness, , for both Type Iand Type II waveguides. In the Type I case we assume a GaAscore with and an Al0.45Ga0.55As cladding with

. Note that for Type II single-mode waveguides,should be 0.3, whereas could be much larger for

Type I waveguides. Roughly speaking, for Type Iwaveguides and for Type II single-mode waveguides.For Type I waveguides, depends also on the waveguidewidth, as shown in Fig. 4(b).

Having solved and , (1) can be reduced to the 2-Dscalar wave equation for

(3)

CHIN AND HO: WAVEGUIDE-COUPLED SINGLE-MODE MICRORING RESONATORS 1435

Fig. 3. Percentage error in effective index calculated by the effective indexmethod and the finite-difference numerical method, for TMoo mode in Type Iand Type II waveguides as a function of waveguide width (a). The waveguidethickness(d) is assumed to be 0.45�m for Type I waveguides. Type IIwaveguides are assumed to be square(a = b).

where . We assume the ring to have a widthof , an inner rim radius of , and an outer rim radius of

, as shown in Fig. 5(a). We next make a transformation to anew coordinate system such that the curved waveguidesbecome straight in the – plane. Such a transformation isgiven by [10]

(4)

Our transformation is somewhat different from the literaturein that we use instead of (or ), where is theeffective ring radius to be defined later. The resulting waveequation in the space is then

(5)

where

forfor and

(6)

and . The effect of waveguidethickness is now embedded in . The transformed linearwaveguide is shown in Fig. 5(b), with the straight edgesat and . Therefractive index in this waveguide is a function of, asillustrated in Fig. 6(a) for the two possible cases correspondingroughly to the Type I and Type II waveguides. In the formercase, the effective refractive index outside the waveguide

(a)

(b)

Fig. 4. (a)nz , the equivalent index in thez direction for the TE modes, asa function ofd=� for Type I and Type II waveguides. (b)nz as a function ofwaveguide width, for Type I waveguides with the given material compositionsand core thickness.

increases radially as so at some point the index willbe large enough to support oscillatory modes. On the otherhand, for Type II waveguides in which , the fieldis completely evanescent outside the guide and there is noradiation loss.

B. The Effect of Diffraction

In reality, the evanescent tail of the bounded mode willdiffract outside the waveguide. This means that, whichis continuous across the boundaries of the waveguide, mustdecrease monotonically once outside the waveguide since thefield there is not guided. The dependence ofon outsidethe waveguide corresponds to the diffraction effect. Here,we approximately include the effect by assuming that theevanescent wave outside the waveguide diffracts as a Gaussianwave, with the value decreasing with according to the


(a)

(b)

Fig. 5. (a) Geometry of the ring waveguide in thex-y plane. (b) Equiv-alent straight waveguide in the(u; v) transform space;u1 and u2 are thecoordinates of the edges of the ring.

usual expression

(7)

where is the effective mode radius at the waveguideedge. This effective mode radius may be approximated by

, where being the decayconstant of outside the waveguide core. The diffractioneffect changes the refractive index distribution outside thewaveguide, as shown in Fig. 6(b). The change is especiallydrastic for the Type II case. In both cases, the external indexfor now increases monotonically with and exceedsthe waveguide effective index at some “turning” point.Essentially, the effect of diffraction is to move this point closerto the waveguide edge. Beyond this point an oscillatory fieldexists meaning that there will always be some radiation lossthrough nonresonant tunneling.

Diffraction has two opposing effects on the field couplingbetween two adjacent waveguides. First, because isreduced, the modal field will now decay less rapidly outside thering waveguide, and therefore overlap more with the adjacentwaveguide. Second, because the field expands in the verticaldirection the coupling is reduced by the decreased modaloverlap in the vertical direction. Overall, the coupling maybe diminished, especially for the Type II structures where thediffraction effect is strong. In other words, to achieve the samecoupling coefficient in both Type I and Type II structures, therequired gap spacing will have to be even smaller in the TypeII structures, which aggravates the difficulty in realizing these

structures. For these reasons, we will henceforth focus mainlyon the Type I structures.

C. The Physical Meaning of

We expect the radial distribution of the eigenmode in acurved waveguide to be somewhat skewed toward the outeredge . Different parts of the radial wavefunction willpropagate in the (or ) direction with slightly differentpathlengths given by . Therefore, for the radial mode as awhole to remain in phase, the propagation constant must varyalong . This shows that in a curved waveguide theand

components are inherently coupled, and the eigenmodes, ingeneral, cannot be separated neatly intoand components.Nevertheless, to make the solution tractable, we shall assumethe separation of variables and write

(8)

In other words, we assume that thecentroid of the radialintensity distribution, , propagates around the ring withan averagepropagation vector . is then theeffectivering radius, defined as the radial distance to thecentroidof the radial function [11]. The resonance condition in thepropagation direction requires that , where isan integer called the azimuthal mode number, which givesthe number of nodes along the direction. Clearly, and

are inter-dependent and must be solved self-consistently,along with the radial function , which is now given bythe one-dimensional (1-D) wave equation

(9)

where ( asbefore). This equation has a discrete set of solutions which arecharacterized by a radial mode number,, giving the numberof nodes in the direction. Once and are known, theresonance wavelength of the cavity is given by

(10)

where , defined by , is implicitly dependenton . The resonance frequency is then given by

, and the FSR is given by. The dependence of on wavelength is known as

the waveguide dispersion.

D. Solution for the Radial Modes

Equation (9) is similar to the Schrodinger equation ofquantum mechanics, with corresponding to the “energy”eigenvalue and to the “potential.” Theproblem of wave propagation (eigenfunctions) in a waveguide(quantum well) with arbitrary refractive index (potential) pro-file can be solved by the transfer matrix method [12] or theWKB method [13]. Here, we aim for a simpler solution, whichis based on the observation that the index profiles in Fig. 6(a)and (b) are approximately linear inside the waveguide, andalso over most regions of interest outside the waveguide wherethe field is not negligibly small. This linear approximation ofthe index profile is good for strongly confined, single-modewaveguides, except when the radii of curvature are very small


(a) (b)

(c)

Fig. 6. Plots of the “potential”ku(u)2 + k2

vas a function ofu for Type I and Type II waveguides: (a) without considering the diffraction effect and (b)

with diffraction taken into account. (c) Linear approximations of the “potential” (with diffraction included) in various regions, showing the two possible caseswhere the potentials in Region II have opposite slopes, resulting in different solutions(FII) in that region. Various notations used in the text are indicated.

m . Within this linear approximation, (9) can besolved analytically in terms of Airy functions, much like theanalogous problem of a quantum well in the presence of anapplied electric field [14]. The same technique has also beenapplied by Goyalet al. to bent planar waveguides [15]. Webriefly describe the method below.

Consider Region I as shown in Fig. 6(c). In the linearapproximation, we write ,

where and are constants to be determined case by case.Equation (9) becomes

(11)

where is the eigenvalue. It can be shown that, bydefining the dimensionless variable


where , (11) is reduced to the Airy equation

(12)

the general solution for which is a linear combination of Airyfunctions

(13)

Similar parameters , and can be defined for RegionsII and III, where the Airy solutions are labeled byand , respectively. The coefficients’s and ’s arethen found by matching boundary conditions at the twointerfaces (corresponding to ) and .The boundary conditions refer to the continuities ofand

. The final result can beexpressed as a matrix equation relating the field amplitudeson both sides of the waveguide

(14)

The eigenvalue equations are then determined by the physicalrequirement that the wavefunctions decay away from thewaveguide. This leads to two distinct cases, with differenteigenvalue equations.

Case i: As shown in part (i) of Fig. 6(c), the behavior of theAiry functions dictates that , and the eigenvalueequation is

(15)

where

Case ii: As shown in part (ii) of Fig. 6(c), in this case, and the eigenvalue equation is

(16)

where

The primes in these equations denote first derivatives of theAiry functions with respect to . Although strictly speakingthe eigenstates are not bound, the decay time of these states areso long that they can be considered as quasistationary states.

III. RESULTS AND DISCUSSIONS

A. Effective Indexes and Radial Wavefunctions

The solutions to (15) or (16) yield the eigen-indexesand eigenfunctions of all the ring waveguide modes. Theeffective index in a microring waveguide is a function

of the radius (say, ), the waveguide width , the verticalwaveguide structure , and the optical wavelength

. The dependence on the vertical waveguide structure islumped into a single parameter as discussed before. Inmost cases, we are interested in waveguides that have a singletransverse mode.

Fig. 7(a)–(c) shows the changes of for TE modes as thewaveguide parameters , and , respectively, are variedwhile the others are kept constant. One can see thatchangessignificantly only if the rings are small, and thatis typicallyless than 0.3 m for single-transverse-mode waveguides at

m. As the width increases, many more radial modescan exist, each having a different curve.

To illustrate the radial eigenfunctions Fig. 8 shows theintensity profiles of the four radial modes

supported in a microring waveguide with a widthof 0.9 m and a radius of 3 m. The indicated values of

give the positions of the “center of gravity” of thesemodes. Let us consider first the high-order modes. Thesemodes tend to be more concentrated near the inner edge ofthe waveguide, implying that the radial propagation of thesemodes is slower near the inner edge. This fact can be deducedfrom Fig. 6(c), which shows that decreases as the inneredge is approached. Next, we consider the low-order modessuch as and . The centroids of these modes are shiftedtoward the outer edge of the waveguide, as shown by the factthat m. In fact, changesslowly to a microdisk-likewhispering gallerymode as thewaveguide width is increased. The transition from microringmode to microdisk mode is illustrated in Fig. 9, which depictsthe normalized intensity profile of as the width of thewaveguide is increased. Note that hardly touches the innerrim, and its position and profile remain unchanged, as long as

m. As the width is increased, the effective indexincreases and moves up into the triangular region of the

“potential” inside the waveguide [see insets]. Onceis insidethe triangular region, the eigenmode sees a potential similar tothat in a microdisk, and therefore is guided in the same way.

B. Resonance Wavelength and Free Spectral Range

Nanoscale waveguides display significant dispersion andbecause the resonance modes for the small ring waveguides arespaced far apart, the effective index will be different at eachresonance wavelength. Over a small wavelength range, thematerial index dispersion may be neglected in the calculationof . A typical result around m is shown inFig. 10(a). It shows that is essentially a linear, decreasingfunction of the wavelength, with a slope dependent somewhaton the width . Using these linear functions, can bedetermined in terms of the azimuthal mode numberforgiven and , according to (10). These are shownin Fig. 10(b) where several important results can be noted.First, for a fixed , as the waveguide width decreases,each resonance is shifted toward shorter wavelength or higherfrequency. Secondly, the wavelength decreases with the modenumber, and the wavelength spacing, or FSR, decreases as thewavelength decreases. This behavior agrees with experimentalobservations [4].


(a) (b)

(c)

Fig. 7. While keeping the other parameters constant, these figures show how the effective indexnv depends on (a)nz , which is linked to the waveguidethickness (d) through Fig. 4, (b) the ring radiusR2 (for variousnz), and (c) the waveguide widthw, showing hownv varies with the radial wavefunctions.

In Table I, the wavelengths and the FSR for major reso-nances around 1.55m in a ring resonator with the parameters

m, m, and m, are listed andcompared with those calculated using the FDTD numericalsimulations [5]. The good agreement between the two setsof results gives us confidence that our method is reasonablyaccurate. For the 5-m diameter resonators, we note that theFSR can be as large as 50 nm (6 THz), sufficient to coverthe entire 30-nm Erbium amplifier bandwidth. The extremelylarge FSR represents one of the advantages of nano-photonicsemiconductor microcavity resonators.

C. Resonant Waveguide Coupling and Coupling Factor

In a waveguide-coupled microring resonator, the couplinggap size is determined by the amount of coupling required

and the coupling length available. A larger gap is desirable asit increases the fabrication tolerance. For a given couplingcoefficient, the gap size can be enlarged if the couplingdistance is increased. The coupling distance can be increasedby using several configurations. A vertical resonant couplingconfiguration is discussed in Section III-F. Twoplanar con-figurations, as shown in Fig. 11, are discussed here. Themore obvious one is the “racetrack” configuration, where thecoupling distance is approximately the length of the straightsections, which are tangential to the semicircular arcs at bothends. At the transitions between the curved and the straightsections, the mode will change adiabatically between the radialmode in a curved waveguide and the normal mode in a straightwaveguide. As the straight sections are lengthened the radiusof the curved sections must be reduced if the total cavity lengthis to remain constant.


Fig. 8. Intensity profiles (in real space) of the four radial modes supportedin a microring waveguide with a width of 0.9�m and a radius of 3�m. TheRe� values give the positions of the centroids of these modes.

Fig. 11(b) shows the alternative configuration where themicroring is coupled to a parallel curved waveguide over asignificant part of the circumference. Note that the physicalpathlengths in the concentric coupled sections in the twowaveguides are different. In other words, if the coupledwaveguides are identical, the propagating wavefronts in thetwo waveguides will in general be out of phase by

after traveling an angle of. Therefore, to maintainphase matching, the two waveguides must bedifferent in sucha way that . This can be achieved byvarying the width of the outer waveguide. Fig. 12(a) showsthe product as a function of , for two concentricwaveguides spaced 0.3m apart. We see that it is possibleto match the optical pathlengths by choosing a suitable com-bination for the two waveguides, such as (0.4m,

Fig. 9. Normalized intensity profile of the lowest order radial mode in amicroring as the width of the waveguide is increased from 0.15�m to 1.5�m. The two insets show the “potential” profiles and positions of the effectiveindex eigenvalues(k2

v) whenw = 0:3 �m andw = 0:6 �m, respectively.

0.2 m) in the case shown. The matching combinations ofare displayed in Fig. 12(b) for two cases of different

radii and gap spacings. Note that the inner waveguide, whichis required to have a larger value of , will always be widerthan the outer waveguide becauseis an increasing functionof [see Fig. 7(c)].

The coupling coefficient between two parallel waveg-uides, in the weak coupling limit applicable in most cases, isgiven by [16]

(17)

where and denote the radial modes in waveguide 1and 2, respectively, and are their propagation vectors,

is the dielectric perturbation and is the perturbation in[cf., (3)] due to the presence of the other waveguide. In the


(a)

(b)

Fig. 10. (a) Effective index of the microring waveguide as a function ofwavelength (in the vicinity of 1.5�m) for two different waveguide widths. (b)The resonance wavelengths and the corresponding azimuthal mode numbers(m).

case of resonant coupling, since the coupled eigenmodes in theconcentric ring waveguides are moving in phase, the couplingcoefficient between them can be calculated in exactly the sameway as for parallel straight waveguides. (The ring nature ofthe waveguides is built into the radial wavefunctions.)

The coupling coefficients are a function of wavelength. Theresults for m are shown in Fig. 13 as a function ofgap spacing. Note that as the gap spacing is changed, the width

TABLE ICALCULATED RESONANCE WAVELENGTHS FOR FSR

FOR A 5-�m DIAMETER MICROCAVITY RING RESONATOR

This method FDTD method

m �m(nm) FSR (nm) �m(nm) FSR (nm)

25 1615.27 1613.1049.96 50.66

26 1565.31 1562.4449.96 47.56

27 1518.35 1514.8844.23 44.73

28 1474.12 1470.1541.72 42.17

29 1432.30 1427.98

(a)

(b)

Fig. 11. Twoplanar geometries for resonant coupling that may give a largercoupling length: (a) the “race-track” configuration and (b) a microring coupledto parallel curved waveguides. Notice the outer waveguide is narrower thanthe inner waveguide.

of the outer waveguide has to be modified somewhat in orderto maintain phase matching with the inner waveguide (isassumed fixed). Once is determined, the coupling factor

, or the fractional power coupled from one waveguide toanother after a length, is given by . Fig. 14 gives

, in the weak coupling limit, for various coupling lengths


(a)

(b)

Fig. 12. (a) The optical pathlength (the product ofnv and Re� ) in twoconcentric ring waveguides, separated by a 0.4-�m wide gap, as a function oftheir waveguide widths. The inner ring has an outer radius(R12) of 5 �m,while the outer ring has an inner radius(R21) of 5.4�m. The phase-matchedpair (w1; w2) is joined by a horizontal line. (b) Matching combinations ofw1 andw2 for two cases with different values of radius and gap spacing.

and gap sizes. It can be used to design the gap size andcoupling length required to achieve a desired coupling factor.The coupling factor should be larger than the scattering loss perround trip, which is typically 1–2%. In the parallel couplingscheme, the coupling length is typically limited to a quadrant,so m (for m). With thiscoupling length, a coupling factor of 1–2% can be achievedwith a coupling gap between 0.3 and 0.4m, or 3–4 times

Fig. 13. The coupling coefficient (in�m�1) between two resonant con-centric ring waveguides as a function of their gap spacing, and the corre-sponding width(w2) required of the outer waveguide in order to maintainphase-matching. The inner waveguide has a fixed width(w1) and radius(R12).

wider than the 0.1-m gap currently used in the tangentialwaveguide structure [4]. Although the fabrication tolerancefor the 0.3- m gap is still small, it is not as stringent as forthe 0.1- m case. For larger microring resonators, the gap willbe even larger and the coupling coefficient will be much lesssensitive to small variations in the gap size.

D. Radiation Loss

The radial field decays as it exits the ring edge until theouter index grows exponentially and becomes greater than theazimuthal index . At this point, becomes real, andthe field begins to propagate once again. This propagation ofthe radial field is a nonresonant coupling of the field to theouter high-index region and constitutes the radiative bendingloss. The value of the field intensity at this point where thedecaying field begins to propagate is analogous to solvingfor the tunneling of light through a lossy barrier, and canbe calculated in a similar fashion to the quantum mechanicaltunneling (such as the alpha-particle emission from a nucleus).In the WKB approximation, the transmission coefficientthrough the barrier (the region from to ) is given by [17]

(18)

where

(19)


Fig. 14. A plot of coupling factor(Pc) as a function of coupling length forvarious gap sizes.

is the radial field decay factor outside the waveguide. Theturning point is given by [see Fig. 6(c)], or

,which is an implicit equation for since is a function of

as in (7).To calculate the round-trip radiation loss, we note that in

making one round-trip, the waveguide mode makes a numberof bounces in the radial direction equal to the azimuthal modenumber, . Thus the round trip loss is

, from which it follows that the radiation loss coefficientis as shown in (20) at the bottom of

the page where is the free-space wavelength. The resultsfor ring waveguides with various diameters and a fixed widthand thickness are shown in Fig. 15. Note that the bendingradiation loss is less than 1% for diameters greater than 1

m, but increases exponentially below that. If the waveguidethickness is decreased, the radiation loss will be larger because,first, is smaller, lowering the tunneling barrier, and second,the increased diffraction effect brings the external high-indexregion closer to the edge of the waveguide, thereby increasingthe tunneling probability.

E. Substrate Leakage Loss

Energy in a waveguide can be lost not only laterally butalso vertically into the substrate. For a Type I waveguide,

Fig. 15. The round-trip radiation loss as a function of the ring radius. Theinset is a plot of the radial wavefunction for the caseR2 = 0:6 �m. Theradiation loss is less than 1% for diameters greater than 1�m.

the waveguide core is typically separated from the substrateby a buffer layer of thickness , as shown in Fig. 2(a). Asignificant amount of signal attenuation in a waveguide canoccur owing to leakage to the substrate if the effective indexof the waveguide is below the bulk index of the substrate.Hence, the design of the buffer layer thickness is importantfor minimizing this leakage. A tradeoff is usually necessarybetween a thick buffer to minimize the leakage loss and a thinbuffer to reduce the etching depth.

The leakage loss to the substrate is very similar to theradiation loss of a curved waveguide, and can be calculatedin exactly the same way using the tunneling picture. For thecase where the buffer layer is etched only partially through,diffraction from the mesa into the unetched buffer can also beconsidered in a similar way. In the case of vertical leakagethe TM polarization is expected to experience greater leakageloss due to the larger extent of the electric field in the verticaldirection. (For similar reason, the TE modes will suffer greaterlateral leakage loss.) A narrow waveguide is also expectedto have a larger substrate leakage loss because its effectiveindex is lower. In Fig. 16(a), is calculated as a function ofbuffer thickness for a 0.5-m wide waveguide with a fixedcore thickness of 0.5 m. Note that a fairly thick buffer

m is required for reasonably small leakage loss.The index contrast should be increased if a smaller

(20)


(a)

(b)

Fig. 16. (a) Substrate leakage loss as a function of the buffer thickness(t2)for TE and TM modes. (b) Substrate leakage loss as a function of waveguidethickness, subject to the constraint thatd + t2 = 2 �m. A broad minimumoccurs aroundd = 0:6 �m.

buffer thickness is desired. In cases where one is constrainedto a fixed etching depth, the core and buffer thicknesses arenot independent, and should be optimized together to yieldthe minimum leakage loss. As an example, Fig. 16(b) givesthe leakage loss as a function of the waveguide thicknessunder the constraint m. A broad minimum occursaround m. For very thin waveguides, the leakageloss increases because of the weak confinement.

F. Vertical Resonant Coupling

The lateral coupling geometries discussed in Section III-Bsuffer from several disadvantages. First, in terms of fabrica-tion, the small gap width is difficult to control. Secondly,without phase matching, the coupling length is very short.To overcome these difficulties, a novel alternative is to usethe vertical coupling geometry, as shown schematically inFig. 17. This is basically a stack structure in which twoidentical waveguide layers are grown epitaxially one on top ofanother. The epiwafer is processed in such a way that the lower

Fig. 17. Schematic of the vertical coupling geometry, showing the verticalstack structure with the ring waveguide (WG1) sitting atop the lower couplingwaveguides (WG2). The separation layer determines the coupling coefficient.

waveguide is the coupling waveguide, and the material outsidethe ring waveguide area is etched away to leave only the inputand output waveguides. The deep etching required to formsuch a vertical structure with low loss is not difficult as thereare no small gaps to worry about. In this structure, the couplingbetween the waveguides is in the vertical direction. The cou-pling mechanism is similar to substrate leakage loss discussedin the previous section, except the coupling is phase-matched.The main advantage is that the coupling coefficient can becontrolled accurately during the epitaxial growth process asit is determined primarily by the thickness of the separationlayer, independent of surface scattering or edge diffraction.Since there is no phase mismatch due to path difference, theinteraction length is the maximum possible. Furthermore, itis relatively easy to make electrical contact to such a device.Direct electrooptic modulation of the coupling coefficient canbe achieved by doping the upper waveguide-type, the lowerwaveguide -type, and the separation layer undoped, andapplying an electric field across the undoped separation layer[18]. Alternatively, if electro-optic modulation of the reso-nance frequency is desired, then the ring waveguide should beundoped and the electric field is applied across it. Despite theseadvantages, however, a major difficulty with this structure isits integrability with other devices. Further investigation of thevertical coupling structure will be given elsewhere.

IV. SUMMARY

In summary, we have discussed the design of a numberof important parameters, including the resonance wavelength,the effective ring radius , the coupling factor ,the bending radiation loss and the substrate leakageloss , all of which are important to the operation ofa waveguide-coupled microring resonator. As an example,consider the resonatorfinesse (the ratio of FSR to thebandwidth) or the factor (the ratio of the absolute frequencyto the bandwidth). These two parameters are both importantwhen one is interested in both the FSR and the bandwidth.They are related by

(21)


(a)

(b)

Fig. 18. (a) The finesse, and (b) theQ factor, of the microring resonator asa function of the ring radius for varying scattering loss coefficients, assuminga coupling factor of 1%. A peak in finesse occurs at the optimal radius ofabout 1�m.

Using this relation, the factor may be written in a form thatshows explicitly its dependence on all the parameters discussedabove

(22)

Fig. 19. The finesse of the microring resonator as a function of diameter,-as the coupling factorsPc1 andPc2 are varied.

where and are the reflectivities, given byat each waveguide-to-ring coupling point ( are

the coupling coefficients at these points). The loss terms in theexponent may include other losses such as absorption loss.,the surface scattering loss in the cavity, is not considered inour model but may yet be the dominant source of loss. Fig. 18shows both the finesse and thefor a case of fixedand varying scattering loss coefficient, , and cm 1.In each case the finesse shows a maximum “turning point” atan optimal diameter of about 2m. The finesse decreasesfor larger diameters because of increasing scattering loss, anddecreases for smaller diameters because of increasing radiativebending loss. On the other hand, shows no such turningpoint in the range shown because the factorin (22) is amonotonically increasing function of the ring diameter at afixed wavelength. The maximum finesse at the turning point islimited by the 1-% waveguide-to-cavity coupling. If there waszero coupling and the scattering loss was also zero, then thefinesse would increase indefinitely with the ring radius. On theother hand, an increase in or an introduction of unbalanced

(i.e., not equal to ) will decrease the finesse, asshown in Fig. 19. The control of , via accurate control ofthe gap size, can be achieved by nanofabrication technologyas has been demonstrated for both GaAs and InP materials [4].With the gap-enhancing designs proposed above, this controlmay be more reliably achieved. On the other hand, a good wayto switch the resonator off is to electrooptically unbalance it sothat the input coupling is different from the output coupling,analogous to the transmission of a Fabry–Perot resonatorwhere front and back mirrors are not identical.

The conformal transformation method at the core of ourmodel gives a good physical picture of the guiding andradiation mechanism in the microring cavity. Much of the


physics can be derived by analogy with quantum wells andtunneling through potential barriers. Based on this physicalpicture, we have developed a novel approach for significantlyincreasing the coupling length, and hence the coupling gapsize, by using a phase-matched parallel waveguide having adifferent width than the ring waveguide. An alternative verticalcoupling structure is also proposed in which the control of thecoupling coefficient is built into the material growth process.The resonant coupling between two waveguides with differentwidths has an appropriate analogy with the resonant couplingbetween two quantum wells with different well widths, whichcan be achieved by applying an appropriate uniform electricfield across the wells. Mathematically, the similarity arisesbecause curving the waveguides has the same effect on (4)(within our linear approximation) as the presence of a uniformelectric field does on the Schrodinger equation for a quantumwell [14].

In summary, we have discussed an approximate model forsolving the effective indexes and the eigenmodes of a micro-ring waveguide with a small radius of curvature and a largelateral index contrast. The model takes into account some 3-Deffects such as waveguide dispersion and edge diffraction. Theresonance frequencies and the FSR calculated with this modelagree well with the FDTD method. Once the ring mode profileand the effective index are known, the coupling coefficientand its dependence on gap size can be obtained readily. Themodel provides physical insight and design guidance usefulfor first-cut optimal designs of waveguide-coupled microringresonators before tedious laboratory fabrications are carriedout.

ACKNOWLEDGMENT

The authors would like to thank Dr. D. Rafizadeh for manyuseful discussions.

REFERENCES

[1] K. Oda, S. Suzuki, H. Takahashi, and T. Toba, “An optical FDMdistribution experiment using a high finesse waveguide-type double ringresonaotr,”IEEE Photon. Technol. Lett., vol. 6, pp. 1031–1034, Aug.1994.

[2] R. Orta, P. Savi, R. Tascone, and D. Trinchero, “Synthesis of multiple-ring-resonator filters for optical systems,”IEEE Photon. Technol. Lett.,vol. 7, pp. 1447–1449, Dec. 1995.

[3] S. Suzuki, K. Oda, and Y. Hibino, “Integrated-optic, double-ring res-onators with a wide free spectral range of 100 GHz,”J. LightwaveTechnol., vol. 13, pp. 1766–1771, Aug. 1995.

[4] D. Rafizadeh, J. P. Zhang, S. C. Hagness, A. Taflove, R. C. Tiberio, K.A. Stair, and S. T. Ho, “Waveguide-coupled AlGaAs/GaAs microcavityring and disk resonators with high finesse and 21.6-nm free spectralrange,”Opt. Lett., vol. 22, pp. 1244–1246, Aug. 1997.

[5] S. C. Hagness, D. Rafizadeh, S. T. Ho, and T. Taflove, “FDTD micro-cavity simulations: Nanoscale waveguide-coupled single-mode ring andwhispering-gallery-mode disk resonators,”J. Lightwave Technol., vol.15, pp. 2154–2165, Nov. 1997.

[6] B. Little, G. S. T. Chi, H. Haus, J. Foresi, and J. P. Laine, “Micro-ringresonator channel dropping filters,”J. Lightwave Technol., vol. 15, p.998, 1997.

[7] M. K. Chin, D. Y. Chu, and S. T. Ho, “Estimation of spontaneousemission factor for microdisk lasers via the approximation of whisperinggallery modes,”J. Appl. Phys., vol. 75, pp. 3302–3307, Apr. 1994.

[8] H. Kogelnik, “Theory of optical waveguides,” inGuided Wave Op-toelectronics, T. Tamir, Ed. New York: Springer-Verlag, 1988, pp.69–74.

[9] The multi-grid finite difference solution was performed withSelene Profrom BBV Software, Enschede, The Netherlands.

[10] M. Heiblum and J. H. Harris, “Analysis of curved optical waveguidesby conformal transformation,”IEEE J. Quant. Electron., vol. 48, pp.2071–2012, Sept. 1969.

[11] D. R. Rowland and J. D. Love, “Evanescent wave coupling of whisper-ing gallery modes of a dielectric cylinder,”Inst. Elect. Eng. Proc.—J.,vol. 140, pp. 177–188, June 1993.

[12] D. Rafizadeh and S. T. Ho, “Numerical analysis of vectorial wavepropagation in waveguides with arbitrary refractive index profiles,”Opt.Commun., vol. 141, pp. 17–20, Aug. 1997.

[13] S. L. McCall, A. F. J. Levi, R. E. Slusher, S. J. Pearton, and R. A.Logan, “Whispering-gallery mode microdisk lasers,”Appl. Phys. Lett.,vol. 60, p. 289, 1992.

[14] M. K. Chin, “Modeling of InGaAs/InAlAs coupled double quantumwells,” J. Appl. Phys., vol. 76, pp. 518–523, 1994.

[15] I. C. Goyal, R. L. Gallawa, and A. K. Ghatak, “Bent planar waveguidesand whispering gallery modes: A new method of analysis,”J. LightwaveTechnol., vol. 8, pp. 768–773, 1990.

[16] A. Yariv and P. Yeh,Optical Waves in Crystals. New York: Wiley,1984, p. 183.

[17] R. L. Liboff, Introductory Quantum Mechanics. New York: Holden-Day, pp. 248–249, 1980.

[18] M. K. Chin, “Design considerations for vertical directional couplers,”J. Lightwave Technol., vol. 11, pp. 1331–1336, 1993.

M. K. Chin (M’92) received the B.S. degree from the Massachusetts Instituteof Technology, Cambridge, in 1986 and the Ph.D. degree from the Universityof California, San Diego, in 1992. His Ph.D. work was on the design andfabrication optimization of quantum-well electroabsorption modulators.

From September 1992 to May 1993, and from March 1997 to June1998, he was a Research Associate at Northwestern University, Evanston,IL, where he worked with Prof. S. T. Ho on microcavity devices andnanofabrications. During the interim, he was an Assistant Professor at theNanyang Technological University, Singapore, where he founded the firstPhotonics Laboratory and did research in soliton communications and diode-pumped solid-state lasers. He is now Manager for Technology Developmentat the United States Integrated Optics, Inc., Evanston, IL, a high-tech start-up.

S. T. Ho (S’83–M’89) received the B.S., M.S., and Ph.D. degrees in electricalengineering from the Massachusetts Institute of Technology, Cambridge, in1984 and 1989, respectively.

From 1989 to 1992, he was a Member of Technical Staff at AT&T BellLaboratories, Murray Hill, NJ. Since 1991, he has been a Faculty Memberin the Department of Electrical and Computer Engineering at NorthwesternUniversity, Evanston, IL. His research areas include microcavity lasers,quantum phenomena in low-dimensional photonic structures, nanofabrication,ultrafast nonlinear optical phenomena, optical communications, and quantumoptics.

Dr. Ho is a member of Phi Beta Kappa, Tau Beta Pi, Sigma Pi Sigma,and Sigma Xi.

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